1 | version="$Id: control.lib,v 1.26 2005-03-17 18:42:48 levandov Exp $"; |
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2 | category="Applications"; |
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3 | info=" |
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4 | LIBRARY: control.lib Procedures for System and Control Theory |
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5 | AUTHORS: Oleksandr Iena yena@mathematik.uni-kl.de |
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6 | @* Markus Becker mbecker@mathematik.uni-kl.de |
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7 | @* Viktor Levandovskyy levandov@mathematik.uni-kl.de |
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8 | |
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9 | SUPPORT: Forschungsschwerpunkt 'Mathematik und Praxis' (Project of Dr. E. Zerz |
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10 | and V. Levandovskyy), Uni Kaiserslautern |
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11 | |
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12 | NOTE: This library provides algebraic analysis tools for System and Control Theory |
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13 | |
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14 | PROCEDURES: |
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15 | control (R); analysis of controllability-related properties of R (using Ext modules), |
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16 | control2 (R); analysis of controllability-related properties of R (using dimension), |
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17 | autonom (R); analysis of autonomy-related properties of R (using Ext modules), |
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18 | autonom2 (R); analysis of autonomy-related properties of R (using dimension), |
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19 | LeftKernel (R); a left kernel of R, |
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20 | RightKernel (R); a right kernel of R, |
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21 | LeftInverse (R); a left inverse of R, |
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22 | RightInverse (R); a right inverse of R, |
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23 | smith (M); a Smith form of a module M, |
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24 | rank (M); a column rank of M as of matrix, |
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25 | genericity (M); analysis of the genericity of parameters, |
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26 | canonize (L); Groebnerification for modules in the output of control/autonomy procs, |
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27 | FindTorsion (R, I); generators of the submodule of a module R, annihilated by the ideal I. |
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28 | |
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29 | |
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30 | AUXILIARY PROCEDURES: |
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31 | declare(NameOfRing,Variables [,Parameters, Ordering]); define the ring easily |
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32 | view(); well-formatted output of lists, modules and matrices |
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33 | |
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34 | NOTE (EXAMPLES): In order to use examples below, execute the commands |
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35 | @* def A = exAntenna(); setring A; |
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36 | @* Thus A will become a basering from the example with the predefined module R (transposed), corresponding to the system. After that you can just type in |
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37 | @* control(R); //respectively autonom(R); |
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38 | and check the result. |
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39 | |
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40 | |
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41 | EXAMPLES (AUTONOMY): |
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42 | exCauchy(); example of 1-dimensional Cauchy equation, |
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43 | exCauchy2(); example of 2-dimensional Cauchy equation, |
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44 | exZerz(); example from the lecture of Eva Zerz, |
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45 | |
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46 | EXAMPLES (CONTROLLABILITY): |
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47 | ex1(); example of noncontrollable system, |
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48 | ex2(); example of controllable system , |
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49 | exAntenna(); Antenna, |
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50 | exEinstein(); Einstein equations, |
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51 | exFlexibleRod(); Flexible Rod, |
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52 | exTwoPendula(); Two Pendula mounted on a cart, |
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53 | exWindTunnel(); Wind Tunnel. |
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54 | "; |
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55 | |
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56 | // NOTE: static things should not be shown for end-user |
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57 | // static Ext_Our(...) Copy of Ext_R from 'homolog.lib' in commutative case; |
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58 | // static is_zero_Our(module R) Copy of is_zero from 'poly.lib'; |
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59 | // static space(int n) Procedure used inside the procedure 'Print' to have a better formatted output |
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60 | // static control_output(); Generating the output for the procedure 'control' |
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61 | // static autonom_output(); Generating the output for the procedure 'autonom' and 'autonom2' |
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62 | |
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63 | LIB "homolog.lib"; |
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64 | LIB "poly.lib"; |
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65 | LIB "primdec.lib"; |
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66 | LIB "matrix.lib"; |
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67 | |
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68 | //--------------------------------------------------------------- |
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69 | static proc Opt_Our() |
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70 | "USAGE: Opt_Our(); |
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71 | RETURN: intvec, where previous options are stored |
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72 | PURPOSE: save previous options and set customized options |
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73 | " |
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74 | { |
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75 | intvec v; |
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76 | v=option(get); |
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77 | option(redSB); |
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78 | option(redTail); |
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79 | return (v); |
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80 | } |
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81 | //--------------------------------------------------------------- |
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82 | proc declare(string NameOfRing, string Variables, list #) |
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83 | "USAGE: declare(NameOfRing, Variables,[Parameters, Ordering]); |
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84 | @* NameOfRing: string with name of ring, |
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85 | @* Variables: string with names of variables separated by commas(e.g. \"x,y,z\"), |
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86 | @* Parameters: string of parameters in the ring separated by commas(e.g. \"a,b,c\"), |
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87 | @* Ordering: string with name of ordering (by default, the ordering \"dp,C\" is used) |
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88 | PURPOSE: define the ring easily |
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89 | RETURN: no return value |
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90 | EXAMPLE: example declare; shows an example |
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91 | " |
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92 | { |
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93 | if(size(#)==0) |
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94 | { |
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95 | execute("ring "+NameOfRing+"=0,("+Variables+"),dp;"); |
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96 | } |
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97 | else |
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98 | { |
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99 | if(size(#)==1) |
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100 | { |
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101 | execute( "ring " + NameOfRing + "=(0," + #[1] + "),(" + Variables + "),dp;" ); |
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102 | } |
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103 | else |
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104 | { |
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105 | if( (size(#[1])!=0)&&(#[1]!=" ") ) |
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106 | { |
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107 | execute( "ring " + NameOfRing + "=(0," + #[1] + "),(" + Variables + "),("+#[2]+");" ); |
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108 | } |
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109 | else |
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110 | { |
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111 | execute( "ring " + NameOfRing + "=0,("+Variables+"),("+#[2]+");" ); |
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112 | }; |
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113 | }; |
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114 | }; |
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115 | keepring(basering); |
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116 | } |
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117 | example |
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118 | {"EXAMPLE:";echo = 2; |
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119 | string v="x,y,z"; |
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120 | string p="q,p"; |
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121 | string Ord ="c,lp"; |
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122 | |
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123 | declare("Ring_1",v); |
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124 | print(nameof(basering)); |
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125 | print(basering); |
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126 | |
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127 | declare("Ring_2",v,p); |
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128 | print(basering); |
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129 | print(nameof(basering)); |
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130 | |
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131 | declare("Ring_3",v,p,Ord); |
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132 | print(basering); |
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133 | print(nameof(basering)); |
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134 | |
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135 | declare("Ring_4",v,"",Ord); |
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136 | print(basering); |
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137 | print(nameof(basering)); |
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138 | |
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139 | declare("Ring_5",v," ",Ord); |
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140 | print(basering); |
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141 | print(nameof(basering)); |
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142 | }; |
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143 | // |
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144 | //maybe reasonable to add this in declare |
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145 | // |
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146 | // print("Please enter your representation matrix in the following form: |
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147 | // module R=[1st row],[2nd row],..."); |
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148 | // print("Type the command: R=transpose(R)"); |
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149 | // print(" To compute controllability please enter: control(R)"); |
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150 | // print(" To compute autonomy please enter: autonom(R)"); |
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151 | //------------------------------------------------------------------------- |
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152 | |
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153 | static proc space(int n) |
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154 | "USAGE:spase(n); |
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155 | n: integer, number of needed spaces |
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156 | RETURN: string consisting of n spaces |
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157 | NOTE: the procedure is used in the procedure 'view' to have a better formatted output |
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158 | " |
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159 | { |
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160 | int i; |
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161 | string s=""; |
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162 | for(i=1;i<=n;i++) |
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163 | { |
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164 | s=s+" "; |
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165 | }; |
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166 | return(s); |
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167 | }; |
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168 | //----------------------------------------------------------------------------- |
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169 | proc view(M) |
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170 | "USAGE: view(M); |
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171 | M: any type |
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172 | RETURN: no return value |
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173 | PURPOSE: procedure for ( well-) formatted output of modules, matrices, lists of modules, matrices; |
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174 | shows everything even if entries are long |
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175 | NOTE: in case of other types( not 'module', 'matrix', 'list') works just as standard 'print' procedure |
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176 | EXAMPLE: example view; shows an example |
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177 | " |
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178 | { |
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179 | // to be replaced with something more feasible |
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180 | if ( (typeof(M)=="module")||(typeof(M)=="matrix") ) |
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181 | { |
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182 | int @R=nrows(M); |
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183 | int @C=ncols(M); |
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184 | int i; |
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185 | int j; |
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186 | list MaxLength=list(); |
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187 | int Size=0; |
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188 | int max; |
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189 | string s; |
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190 | |
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191 | for(i=1;i<=@C;i++) |
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192 | { |
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193 | max=0; |
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194 | |
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195 | for(j=1;j<=@R;j++) |
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196 | { |
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197 | Size=size( string( M[j,i] ) ); |
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198 | if( Size>max ) |
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199 | { |
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200 | max=Size; |
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201 | }; |
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202 | }; |
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203 | MaxLength[i] = max; |
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204 | }; |
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205 | |
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206 | for(i=1;i<=@R;i++) |
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207 | { |
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208 | s=""; |
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209 | for(j=1;j<@C;j++) |
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210 | { |
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211 | s=s+string(M[i,j])+space( MaxLength[j]-size( string( M[i,j] ) ) ) +","; |
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212 | }; |
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213 | |
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214 | s=s+string(M[i,j])+space( MaxLength[j]-size( string( M[i,j] ) ) ); |
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215 | |
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216 | if (i!=@R) |
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217 | { |
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218 | s=s+","; |
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219 | }; |
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220 | print(s); |
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221 | }; |
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222 | |
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223 | return(); |
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224 | }; |
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225 | |
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226 | if(typeof(M)=="list") |
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227 | { |
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228 | int sz=size(M); |
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229 | int i; |
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230 | for(i=1;i<=sz;i++) |
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231 | { |
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232 | view(M[i]); |
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233 | print(""); |
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234 | }; |
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235 | |
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236 | return(); |
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237 | }; |
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238 | print(M); |
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239 | return(); |
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240 | } |
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241 | example |
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242 | {"EXAMPLE:";echo = 2; |
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243 | ring r; |
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244 | matrix M[1][3] = x,y,z; |
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245 | print(M); |
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246 | view(M); |
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247 | }; |
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248 | //-------------------------------------------------------------------------- |
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249 | proc RightKernel(matrix M) |
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250 | "USAGE: RightKernel(M); M a matrix |
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251 | PURPOSE: computes the right kernel of matrix M (a module of all elements v such that Mv=0) |
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252 | RETURN: module |
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253 | EXAMPLE: example RightKernel; shows an example |
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254 | " |
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255 | { |
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256 | return(modulo(M,std(0))); |
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257 | } |
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258 | example |
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259 | {"EXAMPLE:";echo = 2; |
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260 | ring r = 0,(x,y,z),dp; |
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261 | matrix M[1][3] = x,y,z; |
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262 | print(M); |
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263 | matrix R = RightKernel(M); |
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264 | print(R); |
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265 | // check: |
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266 | print(M*R); |
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267 | }; |
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268 | //------------------------------------------------------------------------- |
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269 | proc LeftKernel(matrix M) |
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270 | "USAGE: LeftKernel(M); M a matrix |
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271 | PURPOSE: computes left kernel of matrix M (a module of all elements v such that vM=0) |
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272 | RETURN: module |
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273 | EXAMPLE: example LeftKernel; shows an example |
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274 | " |
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275 | { |
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276 | return( transpose( modulo( transpose(M),std(0) ) ) ); |
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277 | } |
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278 | example |
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279 | {"EXAMPLE:";echo = 2; |
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280 | ring r= 0,(x,y,z),dp; |
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281 | matrix M[3][1] = x,y,z; |
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282 | print(M); |
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283 | matrix L = LeftKernel(M); |
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284 | print(L); |
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285 | // check: |
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286 | print(L*M); |
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287 | }; |
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288 | //------------------------------------------------------------------------ |
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289 | proc LeftInverse(module M) |
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290 | "USAGE: LeftInverse(M); M a module |
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291 | PURPOSE: computes such a matrix L, that LM == Id; |
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292 | RETURN: module |
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293 | EXAMPLE: example LeftInverse; shows an example |
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294 | NOTE: exists only in the case when Id \subseteq M! |
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295 | " |
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296 | { |
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297 | // it works also for the NC case; |
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298 | int NCols = ncols(M); |
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299 | module Id = freemodule(NCols); |
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300 | module N = transpose(M); |
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301 | intvec old_opt=Opt_Our(); |
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302 | Id = std(Id); |
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303 | matrix T; |
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304 | // check the correctness (Id \subseteq M) |
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305 | // via dimension: dim (M) = -1! |
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306 | int d = dim_Our(N); |
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307 | if (d != -1) |
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308 | { |
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309 | // No left inverse exists |
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310 | return(matrix(0)); |
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311 | } |
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312 | matrix T2 = lift(N, Id); |
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313 | T2 = transpose(T2); |
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314 | option(set,old_opt); // set the options back |
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315 | return(T2); |
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316 | } |
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317 | example |
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318 | { // trivial example: |
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319 | "EXAMPLE:";echo =2; |
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320 | ring r = 0,(x,z),dp; |
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321 | matrix M[2][1] = 1,x2z; |
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322 | print(M); |
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323 | print( LeftInverse(M) ); |
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324 | kill r; |
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325 | // derived from the exTwoPendula |
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326 | ring r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
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327 | matrix U[3][1]; |
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328 | U[1,1]=(-L2)*Dt^4+(g)*Dt^2; |
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329 | U[2,1]=(-L1)*Dt^4+(g)*Dt^2; |
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330 | U[3,1]=(L1*L2)*Dt^4+(-g*L1-g*L2)*Dt^2+(g^2); |
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331 | module M = module(U); |
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332 | module L = LeftInverse(M); |
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333 | print(L); |
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334 | }; |
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335 | //----------------------------------------------------------------------- |
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336 | proc RightInverse(module R) |
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337 | "USAGE: RightInverse(M); M a module |
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338 | PURPOSE: computes such a matrix L, that ML == Id; |
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339 | RETURN: module |
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340 | EXAMPLE: example RightInverse; shows an example |
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341 | NOTE: exists only in the case when Id \subseteq M! |
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342 | " |
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343 | { |
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344 | return(transpose(LeftInverse(transpose(R)))); |
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345 | } |
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346 | example |
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347 | { "EXAMPLE:";echo =2; |
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348 | // trivial example: |
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349 | ring r = 0,(x,z),dp; |
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350 | matrix M[1][2] = 1,x2+z; |
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351 | print(M); |
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352 | print( RightInverse(M) ); |
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353 | kill r; |
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354 | // derived from the exTwoPendula |
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355 | ring r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
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356 | matrix U[1][3]; |
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357 | U[1,1]=(-L2)*Dt^4+(g)*Dt^2; |
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358 | U[1,2]=(-L1)*Dt^4+(g)*Dt^2; |
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359 | U[1,3]=(L1*L2)*Dt^4+(-g*L1-g*L2)*Dt^2+(g^2); |
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360 | module M = module(U); |
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361 | module L = RightInverse(M); |
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362 | print(L); |
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363 | }; |
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364 | //----------------------------------------------------------------------- |
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365 | static proc dim_Our(module R) |
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366 | { |
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367 | int d; |
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368 | if (attrib(R,"isSB")<>1) |
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369 | { |
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370 | R = std(R); |
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371 | } |
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372 | d = dim(R); |
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373 | return(d); |
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374 | } |
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375 | //----------------------------------------------------------------------- |
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376 | static proc Ann_Our(module R) |
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377 | { |
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378 | return(Ann(R)); |
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379 | } |
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380 | //----------------------------------------------------------------------- |
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381 | static proc Ext_Our(int i, module R, list #) |
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382 | { |
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383 | // mimicking 'Ext_R' from homolog.lib |
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384 | int ExtraArg = ( size(#)>0 ); |
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385 | if (ExtraArg) |
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386 | { |
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387 | return( Ext_R(i,R,#[1]) ); |
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388 | } |
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389 | else |
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390 | { |
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391 | return( Ext_R(i,R) ); |
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392 | } |
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393 | } |
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394 | //------------------------------------------------------------------------ |
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395 | static proc is_zero_Our |
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396 | { |
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397 | //just a copy of 'is_zero' from "poly.lib" patched with GKdim |
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398 | int d = dim_Our(std(#[1])); |
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399 | int a = ( d==-1 ); |
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400 | if( size(#) >1 ) { list L=a,d; return(L); } |
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401 | return(a); |
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402 | // return( is_zero(R) ) ; |
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403 | }; |
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404 | //------------------------------------------------------------------------ |
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405 | static proc control_output(int i, int NVars, module R, module Ext_1, list Gen) |
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406 | //static proc control_output(int i, int NVars, module R, module Ext_1) |
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407 | "USAGE: control_output(i, NVars, R, Ext_1), where |
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408 | i: integer, number of first nonzero Ext or |
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409 | just a number of variables in a base ring + 1 in case that all the Exts are zero, |
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410 | NVars: integer, number of variables in a base ring, |
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411 | R: module R (cokernel representation), |
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412 | Ext_1: module, the first Ext(its cokernel representation) |
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413 | RETURN: list with all the contollability properties of the system which is to be returned in 'control' procedure |
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414 | NOTE: this procedure is used in 'control' procedure |
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415 | " |
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416 | { |
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417 | // TODO: NVars to be replaced with the global hom. dimension of basering!!! |
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418 | // Is not clear what to do with gl.dim of qrings |
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419 | string DofS = "dimension of the system:"; |
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420 | string Fn = "number of first nonzero Ext:"; |
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421 | string Gen_mes = "Parameter constellations which might lead to a non-controllable system:"; |
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422 | |
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423 | module RK = RightKernel(R); |
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424 | int d=dim_Our(std(transpose(R))); |
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425 | |
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426 | if (i==1) |
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427 | { |
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428 | return( |
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429 | list ( Fn, |
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430 | i, |
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431 | "not controllable , image representation for controllable part:", |
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432 | RK, |
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433 | "kernel representation for controllable part:", |
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434 | LeftKernel( RK ), |
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435 | "obstruction to controllability", |
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436 | Ext_1, |
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437 | "annihilator of torsion module (of obstruction to controllability)", |
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438 | Ann_Our(Ext_1), |
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439 | DofS, |
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440 | d |
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441 | ) |
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442 | ); |
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443 | }; |
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444 | |
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445 | if(i>NVars) |
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446 | { |
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447 | return( list( Fn, |
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448 | -1, |
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449 | "strongly controllable(flat), image representation:", |
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450 | RK, |
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451 | "left inverse to image representation:", |
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452 | LeftInverse(RK), |
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453 | DofS, |
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454 | d, |
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455 | Gen_mes, |
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456 | Gen) |
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457 | ); |
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458 | }; |
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459 | |
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460 | // |
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461 | //now i<=NVars |
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462 | // |
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463 | |
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464 | if( (i==2) ) |
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465 | { |
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466 | return( list( Fn, |
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467 | i, |
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468 | "controllable, not reflexive, image representation:", |
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469 | RK, |
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470 | DofS, |
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471 | d, |
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472 | Gen_mes, |
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473 | Gen) |
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474 | ); |
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475 | }; |
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476 | |
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477 | if( (i>=3) ) |
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478 | { |
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479 | return( list ( Fn, |
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480 | i, |
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481 | "reflexive, not strongly controllable, image representation:", |
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482 | RK, |
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483 | DofS, |
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484 | d, |
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485 | Gen_mes, |
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486 | Gen) |
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487 | ); |
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488 | }; |
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489 | }; |
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490 | //------------------------------------------------------------------------- |
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491 | |
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492 | proc control(module R) |
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493 | "USAGE: control(R); R a module (R is the matrix of the system of equations to be investigated) |
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494 | PURPOSE: compute the list of all the properties concerning controllability of the system (behavior), represented by the matrix R |
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495 | RETURN: list |
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496 | EXAMPLE: example control; shows an example |
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497 | " |
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498 | { |
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499 | int i; |
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500 | int NVars=nvars(basering); |
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501 | // TODO: NVars to be replaced with the global hom. dimension of basering!!! |
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502 | int ExtIsZero; |
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503 | intvec v=Opt_Our(); |
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504 | module R_std=std(R); |
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505 | module Ext_1 = std(Ext_Our(1,R_std)); |
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506 | |
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507 | ExtIsZero=is_zero_Our(Ext_1); |
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508 | i=1; |
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509 | while( (ExtIsZero) && (i<=NVars) ) |
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510 | { |
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511 | i++; |
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512 | ExtIsZero = is_zero_Our( Ext_Our(i,R_std) ); |
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513 | }; |
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514 | matrix T=lift(R,R_std); |
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515 | list l=genericity(T); |
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516 | option(set,v); |
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517 | |
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518 | return( control_output( i, NVars, R, Ext_1, l ) ); |
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519 | } |
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520 | example |
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521 | {"EXAMPLE:";echo = 2; |
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522 | //Wind Tunnel |
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523 | ring A = (0,a, omega, zeta, k),(D1, delta),dp; |
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524 | module R; |
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525 | R = [D1+a, -k*a*delta, 0, 0], |
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526 | [0, D1, -1, 0], |
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527 | [0, omega^2, D1+2*zeta*omega, -omega^2]; |
---|
528 | R=transpose(R); |
---|
529 | view(R); |
---|
530 | view(control(R)); |
---|
531 | }; |
---|
532 | //-------------------------------------------------------------------------- |
---|
533 | proc control2(module R) |
---|
534 | "USAGE: control2(R); R a module (R is the matrix of the system of equations to be investigated) |
---|
535 | PURPOSE: computes list of all the properties concerning controllability of the system (behavior), represented by the matrix R |
---|
536 | RETURN: list |
---|
537 | EXAMPLE: example control2; shows an example |
---|
538 | NOTE: same as control(R); but using dimensions, this procedure only works for full row rank matrices |
---|
539 | " |
---|
540 | { |
---|
541 | if( nrows(R) != rank(transpose(R)) ) |
---|
542 | { |
---|
543 | return ("control2 cannot be applied, since R does not have full row rank"); |
---|
544 | } |
---|
545 | intvec v=option(get); |
---|
546 | module R_std=std(R); |
---|
547 | int d=dim_Our(R_std); |
---|
548 | int NVars=nvars(basering); |
---|
549 | int i=NVars-d; |
---|
550 | module Ext_1; |
---|
551 | |
---|
552 | if(i==1) |
---|
553 | { |
---|
554 | Ext_1=std(Ext_Our(1,R_std)); |
---|
555 | } |
---|
556 | matrix T=lift(R,R_std); |
---|
557 | list l=genericity(T); |
---|
558 | option(set, v); |
---|
559 | return( control_output( i, NVars, R, Ext_1, l)); |
---|
560 | } |
---|
561 | example |
---|
562 | {"EXAMPLE:";echo = 2; |
---|
563 | //Wind Tunnel |
---|
564 | ring A = (0,a, omega, zeta, k),(D1, delta),dp; |
---|
565 | module R; |
---|
566 | R = [D1+a, -k*a*delta, 0, 0], |
---|
567 | [0, D1, -1, 0], |
---|
568 | [0, omega^2, D1+2*zeta*omega, -omega^2]; |
---|
569 | R=transpose(R); |
---|
570 | view(R); |
---|
571 | view(control2(R)); |
---|
572 | }; |
---|
573 | //------------------------------------------------------------------------ |
---|
574 | proc rank(module M) |
---|
575 | "USAGE: proc rank(M), M a matrix/module |
---|
576 | PURPOSE: compute the column rank of M as of matrix |
---|
577 | RETURN: int |
---|
578 | NOTE: this procedure uses bareiss-algorithm which might not terminate |
---|
579 | " |
---|
580 | { |
---|
581 | module M_red = bareiss(M)[1]; |
---|
582 | int NCols_red = ncols(M_red); |
---|
583 | return (NCols_red); |
---|
584 | } |
---|
585 | example |
---|
586 | {"EXAMPLE: ";echo = 2; |
---|
587 | // de Rham complex |
---|
588 | ring r=0,(D(1..3)),dp; |
---|
589 | module R; |
---|
590 | R=[0,-D(3),D(2)], |
---|
591 | [D(3),0,-D(1)], |
---|
592 | [-D(2),D(1),0]; |
---|
593 | R=transpose(R); |
---|
594 | rank(R); |
---|
595 | }; |
---|
596 | |
---|
597 | //------------------------------------------------------------------------ |
---|
598 | static proc autonom_output( int i, int NVars, module RC, int R_rank ) |
---|
599 | "USAGE: proc autonom_output(i, NVars, RC, R_rank) |
---|
600 | i: integer, number of first nonzero Ext or |
---|
601 | just number of variables in a base ring + 1 in case that all the Exts are zero |
---|
602 | NVars: integer, number of variables in a base ring |
---|
603 | RC: module, kernel-representation of controllable part of the system |
---|
604 | R_rank: integer, rank of the representation matrix |
---|
605 | PURPOSE: compute all the autonomy properties of the system which is to be returned in 'autonom' procedure |
---|
606 | RETURN: list |
---|
607 | NOTE: this procedure is used in 'autonom' procedure |
---|
608 | " |
---|
609 | { |
---|
610 | int d=NVars-i;//that is the dimension of the system |
---|
611 | string DofS="dimension of the system:"; |
---|
612 | string Fn = "number of first nonzero Ext:"; |
---|
613 | if(i==0) |
---|
614 | { |
---|
615 | return( list( Fn, |
---|
616 | i, |
---|
617 | "not autonomous", |
---|
618 | "kernel representation for controllable part", |
---|
619 | RC, |
---|
620 | "column-rank of the matrix", |
---|
621 | R_rank, |
---|
622 | DofS, |
---|
623 | d ) |
---|
624 | ); |
---|
625 | }; |
---|
626 | |
---|
627 | if( i>NVars ) |
---|
628 | { |
---|
629 | return( list( Fn, |
---|
630 | -1, |
---|
631 | "trivial", |
---|
632 | DofS, |
---|
633 | d ) |
---|
634 | ); |
---|
635 | }; |
---|
636 | |
---|
637 | // |
---|
638 | //now i<=NVars |
---|
639 | // |
---|
640 | |
---|
641 | |
---|
642 | if( i==1 ) |
---|
643 | // in case that NVars==1 there is no sense to consider the notion |
---|
644 | // of strongly autonomous behavior, because it does not imply |
---|
645 | // that system is overdetermined in this case |
---|
646 | { |
---|
647 | return( list ( Fn, |
---|
648 | i, |
---|
649 | "autonomous, not overdetermined", |
---|
650 | DofS, |
---|
651 | d ) |
---|
652 | ); |
---|
653 | }; |
---|
654 | |
---|
655 | if( i==NVars ) |
---|
656 | { |
---|
657 | return( list( Fn, |
---|
658 | i, |
---|
659 | "strongly autonomous(fin. dimensional),in particular overdetermined", |
---|
660 | DofS, |
---|
661 | d) |
---|
662 | ); |
---|
663 | }; |
---|
664 | |
---|
665 | if( i<NVars ) |
---|
666 | { |
---|
667 | return( list ( Fn, |
---|
668 | i, |
---|
669 | "overdetermined, not strongly autonomous", |
---|
670 | DofS, |
---|
671 | d) |
---|
672 | ); |
---|
673 | }; |
---|
674 | }; |
---|
675 | //-------------------------------------------------------------------------- |
---|
676 | proc autonom2(module R) |
---|
677 | "USAGE: autonom2(R); R a module (R is a matrix of the system of equations which is to be investigated) |
---|
678 | PURPOSE: computes the list of all the properties concerning autonomy of the system (behavior), represented by the matrix R |
---|
679 | RETURN: list |
---|
680 | NOTE: this procedure is analogous to 'autonom' but uses dimension calculations |
---|
681 | EXAMPLE: example autonom2; shows an example |
---|
682 | " |
---|
683 | { |
---|
684 | int d; |
---|
685 | int NVars = nvars(basering); |
---|
686 | module RT = transpose(R); |
---|
687 | module RC; //for computation of controllable part if if exists |
---|
688 | int R_rank = ncols(R); |
---|
689 | d = dim_Our( std(RT) ); //this is the dimension of the system |
---|
690 | int i = NVars-d; //First non-zero Ext |
---|
691 | if( d==0 ) |
---|
692 | { |
---|
693 | RC=LeftKernel(RightKernel(R)); |
---|
694 | R_rank=rank(R); |
---|
695 | } |
---|
696 | return( autonom_output(i,NVars,RC,R_rank) ); |
---|
697 | } |
---|
698 | example |
---|
699 | {"EXAMPLE:"; echo = 2; |
---|
700 | //Cauchy |
---|
701 | ring r=0,(s1,s2,s3,s4),dp; |
---|
702 | module R= [s1,-s2], |
---|
703 | [s2, s1], |
---|
704 | [s3,-s4], |
---|
705 | [s4, s3]; |
---|
706 | R=transpose(R); |
---|
707 | view( R ); |
---|
708 | view( autonom2(R) ); |
---|
709 | }; |
---|
710 | //---------------------------------------------------------- |
---|
711 | proc autonom(module R) |
---|
712 | "USAGE: autonom(R); R a module (R is a matrix of the system of equations which is to be investigated) |
---|
713 | PURPOSE: find all the properties concerning autonomy of the system (behavior) represented by the matrix R |
---|
714 | RETURN: list |
---|
715 | EXAMPLE: example autonom; shows an example |
---|
716 | " |
---|
717 | { |
---|
718 | int NVars=nvars(basering); |
---|
719 | int ExtIsZero; |
---|
720 | module RT=transpose(R); |
---|
721 | module RC; |
---|
722 | int R_rank=ncols(R); |
---|
723 | ExtIsZero=is_zero_Our(Ext_Our(0,RT)); |
---|
724 | int i=0; |
---|
725 | while( (ExtIsZero)&&(i<=NVars) ) |
---|
726 | { |
---|
727 | i++; |
---|
728 | ExtIsZero = is_zero_Our(Ext_Our(i,RT)); |
---|
729 | }; |
---|
730 | if (i==0) |
---|
731 | { |
---|
732 | RC=LeftKernel(RightKernel(R)); |
---|
733 | R_rank=rank(R); |
---|
734 | } |
---|
735 | return(autonom_output(i,NVars,RC,R_rank)); |
---|
736 | } |
---|
737 | example |
---|
738 | {"EXAMPLE:"; echo = 2; |
---|
739 | // Cauchy |
---|
740 | ring r=0,(s1,s2,s3,s4),dp; |
---|
741 | module R= [s1,-s2], |
---|
742 | [s2, s1], |
---|
743 | [s3,-s4], |
---|
744 | [s4, s3]; |
---|
745 | R=transpose(R); |
---|
746 | view( R ); |
---|
747 | view( autonom(R) ); |
---|
748 | }; |
---|
749 | |
---|
750 | //---------------------------------------------------------- |
---|
751 | // |
---|
752 | //Some example rings with defined systems |
---|
753 | //---------------------------------------------------------- |
---|
754 | //autonomy: |
---|
755 | // |
---|
756 | //---------------------------------------------------------- |
---|
757 | proc exCauchy() |
---|
758 | { |
---|
759 | ring @r=0,(s1,s2),dp; |
---|
760 | module R= [s1,-s2], |
---|
761 | [s2, s1]; |
---|
762 | R=transpose(R); |
---|
763 | export R; |
---|
764 | return(@r); |
---|
765 | }; |
---|
766 | //---------------------------------------------------------- |
---|
767 | proc exCauchy2() |
---|
768 | { |
---|
769 | ring @r=0,(s1,s2,s3,s4),dp; |
---|
770 | module R= [s1,-s2], |
---|
771 | [s2, s1], |
---|
772 | [s3,-s4], |
---|
773 | [s4, s3]; |
---|
774 | R=transpose(R); |
---|
775 | export R; |
---|
776 | return(@r); |
---|
777 | }; |
---|
778 | //---------------------------------------------------------- |
---|
779 | proc exZerz() |
---|
780 | { |
---|
781 | ring @r=0,(d1,d2),dp; |
---|
782 | module R=[d1^2-d2], |
---|
783 | [d2^2-1]; |
---|
784 | R=transpose(R); |
---|
785 | export R; |
---|
786 | return(@r); |
---|
787 | }; |
---|
788 | //---------------------------------------------------------- |
---|
789 | //control |
---|
790 | //---------------------------------------------------------- |
---|
791 | proc ex1() |
---|
792 | { |
---|
793 | ring @r=0,(s1,s2,s3),dp; |
---|
794 | module R=[0,-s3,s2], |
---|
795 | [s3,0,-s1]; |
---|
796 | R=transpose(R); |
---|
797 | export R; |
---|
798 | return(@r); |
---|
799 | }; |
---|
800 | //---------------------------------------------------------- |
---|
801 | proc ex2() |
---|
802 | { |
---|
803 | ring @r=0,(s1,s2,s3),dp; |
---|
804 | module R=[0,-s3,s2], |
---|
805 | [s3,0,-s1], |
---|
806 | [-s2,s1,0]; |
---|
807 | R=transpose(R); |
---|
808 | export R; |
---|
809 | return(@r); |
---|
810 | }; |
---|
811 | //---------------------------------------------------------- |
---|
812 | proc exAntenna() |
---|
813 | { |
---|
814 | ring @r = (0, K1, K2, Te, Kp, Kc),(Dt, delta), dp; |
---|
815 | module R; |
---|
816 | R = [Dt, -K1, 0, 0, 0, 0, 0, 0, 0], |
---|
817 | [0, Dt+K2/Te, 0, 0, 0, 0, -Kp/Te*delta, -Kc/Te*delta, -Kc/Te*delta], |
---|
818 | [0, 0, Dt, -K1, 0, 0, 0, 0, 0], |
---|
819 | [0, 0, 0, Dt+K2/Te, 0, 0, -Kc/Te*delta, -Kp/Te*delta, -Kc/Te*delta], |
---|
820 | [0, 0, 0, 0, Dt, -K1, 0, 0, 0], |
---|
821 | [0, 0, 0, 0, 0, Dt+K2/Te, -Kc/Te*delta, -Kc/Te*delta, -Kp/Te*delta]; |
---|
822 | |
---|
823 | R=transpose(R); |
---|
824 | export R; |
---|
825 | return(@r); |
---|
826 | }; |
---|
827 | |
---|
828 | //---------------------------------------------------------- |
---|
829 | |
---|
830 | proc exEinstein() |
---|
831 | { |
---|
832 | ring @r = 0,(D(1..4)),dp; |
---|
833 | module R = |
---|
834 | [D(2)^2+D(3)^2-D(4)^2, D(1)^2, D(1)^2, -D(1)^2, -2*D(1)*D(2), 0, 0, -2*D(1)*D(3), 0, 2*D(1)*D(4)], |
---|
835 | [D(2)^2, D(1)^2+D(3)^2-D(4)^2, D(2)^2, -D(2)^2, -2*D(1)*D(2), -2*D(2)*D(3), 0, 0, 2*D(2)*D(4), 0], |
---|
836 | [D(3)^2, D(3)^2, D(1)^2+D(2)^2-D(4)^2, -D(3)^2, 0, -2*D(2)*D(3), 2*D(3)*D(4), -2*D(1)*D(3), 0, 0], |
---|
837 | [D(4)^2, D(4)^2, D(4)^2, D(1)^2+D(2)^2+D(3)^2, 0, 0, -2*D(3)*D(4), 0, -2*D(2)*D(4), -2*D(1)*D(4)], |
---|
838 | [0, 0, D(1)*D(2), -D(1)*D(2), D(3)^2-D(4)^2, -D(1)*D(3), 0, -D(2)*D(3), D(1)*D(4), D(2)*D(4)], |
---|
839 | [D(2)*D(3), 0, 0, -D(2)*D(3),-D(1)*D(3), D(1)^2-D(4)^2, D(2)*D(4), -D(1)*D(2), D(3)*D(4), 0], |
---|
840 | [D(3)*D(4), D(3)*D(4), 0, 0, 0, -D(2)*D(4), D(1)^2+D(2)^2, -D(1)*D(4), -D(2)*D(3), -D(1)*D(3)], |
---|
841 | [0, D(1)*D(3), 0, -D(1)*D(3), -D(2)*D(3), -D(1)*D(2), D(1)*D(4), D(2)^2-D(4)^2, 0, D(3)*D(4)], |
---|
842 | [D(2)*D(4), 0, D(2)*D(4), 0, -D(1)*D(4), -D(3)*D(4), -D(2)*D(3), 0, D(1)^2+D(3)^2, -D(1)*D(2)], |
---|
843 | [0, D(1)*D(4), D(1)*D(4), 0, -D(2)*D(4), 0, -D(1)*D(3), -D(3)*D(4), -D(1)*D(2), D(2)^2+D(3)^2]; |
---|
844 | |
---|
845 | R=transpose(R); |
---|
846 | export R; |
---|
847 | return(@r); |
---|
848 | }; |
---|
849 | |
---|
850 | //---------------------------------------------------------- |
---|
851 | proc exFlexibleRod() |
---|
852 | { |
---|
853 | ring @r = 0,(D1, delta), dp; |
---|
854 | module R; |
---|
855 | R = [D1, -D1*delta, -1], [2*D1*delta, -D1-D1*delta^2, 0]; |
---|
856 | |
---|
857 | R=transpose(R); |
---|
858 | export R; |
---|
859 | return(@r); |
---|
860 | }; |
---|
861 | |
---|
862 | //---------------------------------------------------------- |
---|
863 | proc exTwoPendula() |
---|
864 | { |
---|
865 | ring @r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
---|
866 | module R = [m1*L1*Dt^2, m2*L2*Dt^2, -1, (M+m1+m2)*Dt^2], |
---|
867 | [m1*L1^2*Dt^2-m1*L1*g, 0, 0, m1*L1*Dt^2], |
---|
868 | [0, m2*L2^2*Dt^2-m2*L2*g, 0, m2*L2*Dt^2]; |
---|
869 | |
---|
870 | R=transpose(R); |
---|
871 | export R; |
---|
872 | return(@r); |
---|
873 | }; |
---|
874 | //---------------------------------------------------------- |
---|
875 | proc exWindTunnel() |
---|
876 | { |
---|
877 | ring @r = (0,a, omega, zeta, k),(D1, delta),dp; |
---|
878 | module R = [D1+a, -k*a*delta, 0, 0], |
---|
879 | [0, D1, -1, 0], |
---|
880 | [0, omega^2, D1+2*zeta*omega, -omega^2]; |
---|
881 | |
---|
882 | R=transpose(R); |
---|
883 | export R; |
---|
884 | return(@r); |
---|
885 | }; |
---|
886 | //---------------------------------------------------------- |
---|
887 | proc genericity(matrix M) |
---|
888 | "USAGE: genericity(M), M is a matrix/module |
---|
889 | PURPOSE: determine parametric expressions which have been assumed to be non-zero in the process of computing the Groebner basis |
---|
890 | RETURN: list (of strings) |
---|
891 | NOTE: we strongly recommend to switch on the redSB and redTail options; |
---|
892 | @* the procedure is effective with the lift procedure for modules with parameters |
---|
893 | EXAMPLE: example genericity; shows an example |
---|
894 | " |
---|
895 | { |
---|
896 | // returns "-", if there are no parameters! |
---|
897 | if (npars(basering)==0) |
---|
898 | { |
---|
899 | return("-"); |
---|
900 | } |
---|
901 | list RT = evas_genericity(M); // list of strings |
---|
902 | if ((size(RT)==1) && (RT[1] == "")) |
---|
903 | { |
---|
904 | return("-"); |
---|
905 | } |
---|
906 | return(RT); |
---|
907 | } |
---|
908 | example |
---|
909 | { // TwoPendula |
---|
910 | "EXAMPLE:"; echo = 2; |
---|
911 | ring r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
---|
912 | module RR = |
---|
913 | [m1*L1*Dt^2, m2*L2*Dt^2, -1, (M+m1+m2)*Dt^2], |
---|
914 | [m1*L1^2*Dt^2-m1*L1*g, 0, 0, m1*L1*Dt^2], |
---|
915 | [0, m2*L2^2*Dt^2-m2*L2*g, 0, m2*L2*Dt^2]; |
---|
916 | module R = transpose(RR); |
---|
917 | module SR = std(R); |
---|
918 | matrix T = lift(R,SR); |
---|
919 | genericity(T); |
---|
920 | //-- The result might be different when computing reduced bases: |
---|
921 | matrix T2; |
---|
922 | option(redSB); |
---|
923 | option(redTail); |
---|
924 | module SR2 = std(R); |
---|
925 | T2 = lift(R,SR2); |
---|
926 | genericity(T2); |
---|
927 | } |
---|
928 | //--------------------------------------------------------------- |
---|
929 | static proc victors_genericity(matrix M) |
---|
930 | { |
---|
931 | // returns "-", if there are no parameters! |
---|
932 | if (npars(basering)==0) |
---|
933 | { |
---|
934 | return("-"); |
---|
935 | } |
---|
936 | int plevel = printlevel-voice+2; |
---|
937 | // M is a matrix over a ring with params and vars; |
---|
938 | ideal I = ideal(M); // a list of entries |
---|
939 | I = simplify(I,2); // delete 0's |
---|
940 | // decompose every coeff in every poly |
---|
941 | int i; |
---|
942 | int s = size(I); |
---|
943 | ideal NM; |
---|
944 | poly p; |
---|
945 | number num; |
---|
946 | int cl=1; |
---|
947 | intvec ZeroVec; ZeroVec[nvars(basering)] = 0; |
---|
948 | intvec W; |
---|
949 | ideal Numero, Denomiro; |
---|
950 | int cNu=0; int cDe=0; |
---|
951 | for (i=1; i<=s; i++) |
---|
952 | { |
---|
953 | // remove contents and add them as polys |
---|
954 | p = I[i]; |
---|
955 | W = leadexp(p); |
---|
956 | if (W == ZeroVec) // i.e. just a coef |
---|
957 | { |
---|
958 | num = denominator(leadcoef(p)); // from poly.lib |
---|
959 | NM[cl] = numerator(leadcoef(p)); |
---|
960 | dbprint(p,"numerator:"); |
---|
961 | dbprint(p, string(NM[cl])); |
---|
962 | cNu++; Numero[cNu]= NM[cl]; |
---|
963 | cl++; |
---|
964 | NM[cl] = num; // denominator |
---|
965 | dbprint(p,"denominator:"); |
---|
966 | dbprint(p, string(NM[cl])); |
---|
967 | cDe++; Denomiro[cDe]= NM[cl]; |
---|
968 | cl++; |
---|
969 | p = p - lead(p); // for the next cycle |
---|
970 | } |
---|
971 | if ( p!= 0) |
---|
972 | { |
---|
973 | num = content(p); |
---|
974 | p = p/num; |
---|
975 | NM[cl] = denominator(num); |
---|
976 | dbprint(p,"content denominator:"); |
---|
977 | dbprint(p, string(NM[cl])); |
---|
978 | cNu++; Numero[cNu]= NM[cl]; |
---|
979 | cl++; |
---|
980 | NM[cl] = numerator(num); |
---|
981 | dbprint(p,"content numerator:"); |
---|
982 | dbprint(p, string(NM[cl])); |
---|
983 | cDe++; Denomiro[cDe]= NM[cl]; |
---|
984 | cl++; |
---|
985 | } |
---|
986 | // it seems that the next elements will not have real influence |
---|
987 | while( p != 0) |
---|
988 | { |
---|
989 | NM[cl] = leadcoef(p); // should be all integer, i.e. non-rational |
---|
990 | dbprint(p,"coef:"); |
---|
991 | dbprint(p, string(NM[cl])); |
---|
992 | cl++; |
---|
993 | p = p - lead(p); |
---|
994 | } |
---|
995 | } |
---|
996 | NM = simplify(NM,4); // delete identical |
---|
997 | string newvars = parstr(basering); |
---|
998 | def save = basering; |
---|
999 | string NewRing = "ring @NR =" +string(char(basering))+",("+newvars+"),Dp;"; |
---|
1000 | execute(NewRing); |
---|
1001 | // get params as variables |
---|
1002 | // create a list of non-monomials |
---|
1003 | ideal @L; |
---|
1004 | ideal F; |
---|
1005 | ideal NM = imap(save,NM); |
---|
1006 | NM = simplify(NM,8); //delete multiples |
---|
1007 | poly p,q; |
---|
1008 | cl = 1; |
---|
1009 | int j, cf; |
---|
1010 | for(i=1; i<=size(NM);i++) |
---|
1011 | { |
---|
1012 | p = NM[i] - lead(NM[i]); |
---|
1013 | if (p!=0) |
---|
1014 | { |
---|
1015 | // L[cl] = p; |
---|
1016 | F = factorize(NM[i],1); //non-constant factors only |
---|
1017 | cf = 1; |
---|
1018 | // factorize every polynomial |
---|
1019 | // throw away every monomial from factorization (also constants from above ring) |
---|
1020 | for (j=1; j<=size(F);j++) |
---|
1021 | { |
---|
1022 | q = F[j]-lead(F[j]); |
---|
1023 | if (q!=0) |
---|
1024 | { |
---|
1025 | @L[cl] = F[j]; |
---|
1026 | cl++; |
---|
1027 | } |
---|
1028 | } |
---|
1029 | } |
---|
1030 | } |
---|
1031 | // return the result [in string-format] |
---|
1032 | @L = simplify(@L,2+4+8); // skip zeroes, doubled and entries, diff. by a constant |
---|
1033 | list SL; |
---|
1034 | for (j=1; j<=size(@L);j++) |
---|
1035 | { |
---|
1036 | SL[j] = string(@L[j]); |
---|
1037 | } |
---|
1038 | setring save; |
---|
1039 | return(SL); |
---|
1040 | } |
---|
1041 | //--------------------------------------------------------------- |
---|
1042 | static proc evas_genericity(matrix M) |
---|
1043 | { |
---|
1044 | // called from the main genericity proc |
---|
1045 | ideal I = ideal(M); |
---|
1046 | I = simplify(I,2+4); |
---|
1047 | int s = size(I); |
---|
1048 | ideal Den; |
---|
1049 | poly p; |
---|
1050 | int i; |
---|
1051 | for (i=1; i<=s; i++) |
---|
1052 | { |
---|
1053 | p = I[i]; |
---|
1054 | while (p !=0) |
---|
1055 | { |
---|
1056 | Den = Den, denominator(leadcoef(p)); |
---|
1057 | p = p-lead(p); |
---|
1058 | } |
---|
1059 | } |
---|
1060 | Den = simplify(Den,2+4); |
---|
1061 | string newvars = parstr(basering); |
---|
1062 | def save = basering; |
---|
1063 | string NewRing = "ring @NR =" +string(char(basering))+",("+newvars+"),Dp;"; |
---|
1064 | execute(NewRing); |
---|
1065 | ideal F; |
---|
1066 | ideal Den = imap(save,Den); |
---|
1067 | Den = simplify(Den,2); |
---|
1068 | int s1 = size(Den); |
---|
1069 | for (i=1; i<=s1; i++) |
---|
1070 | { |
---|
1071 | if (Den[i] !=1) |
---|
1072 | { |
---|
1073 | F= F, factorize(Den[i],1); |
---|
1074 | } |
---|
1075 | } |
---|
1076 | F = simplify(F, 2+4+8); |
---|
1077 | ideal @L = F; |
---|
1078 | list SL; |
---|
1079 | int c,j; |
---|
1080 | string Mono; |
---|
1081 | c = 1; |
---|
1082 | for (j=1; j<=size(@L);j++) |
---|
1083 | { |
---|
1084 | if (leadcoef(@L[j]) <0) |
---|
1085 | { |
---|
1086 | @L[j] = -1*@L[j]; |
---|
1087 | } |
---|
1088 | if ( (@L[j] - lead(@L[j]))==0 ) //@L[j] is a monomial |
---|
1089 | { |
---|
1090 | Mono = Mono + string(@L[j])+ ","; // concatenation |
---|
1091 | } |
---|
1092 | else |
---|
1093 | { |
---|
1094 | c++; |
---|
1095 | SL[c] = string(@L[j]); |
---|
1096 | } |
---|
1097 | } |
---|
1098 | if (Mono!="") |
---|
1099 | { |
---|
1100 | Mono = Mono[1..size(Mono)-1]; // delete the last semicolon |
---|
1101 | } |
---|
1102 | SL[1] = Mono; |
---|
1103 | setring save; |
---|
1104 | return(SL); |
---|
1105 | } |
---|
1106 | |
---|
1107 | //--------------------------------------------------------------- |
---|
1108 | proc canonize(list L) |
---|
1109 | "USAGE: canonize(L), L a list |
---|
1110 | PURPOSE: modules in the list are canonized by computing their reduced minimal (= unique up to constant factor w.r.t. the given ordering) Groebner bases |
---|
1111 | RETURN: list |
---|
1112 | ASSUME: L is the output of control/autonomy procedures |
---|
1113 | EXAMPLE: example canonize; shows an example |
---|
1114 | " |
---|
1115 | { |
---|
1116 | list M = L; |
---|
1117 | intvec v=Opt_Our(); |
---|
1118 | int s = size(L); |
---|
1119 | int i; |
---|
1120 | for (i=2; i<=s; i=i+2) |
---|
1121 | { |
---|
1122 | if (typeof(M[i])=="module") |
---|
1123 | { |
---|
1124 | M[i] = std(M[i]); |
---|
1125 | // M[i] = prune(M[i]); // mimimal embedding: no need yet |
---|
1126 | // M[i] = std(M[i]); |
---|
1127 | } |
---|
1128 | } |
---|
1129 | option(set, v); //set old values back |
---|
1130 | return(M); |
---|
1131 | } |
---|
1132 | example |
---|
1133 | { // TwoPendula with L1=L2=L |
---|
1134 | "EXAMPLE:"; echo = 2; |
---|
1135 | ring r=(0,m1,m2,M,g,L),Dt,dp; |
---|
1136 | module RR = |
---|
1137 | [m1*L*Dt^2, m2*L*Dt^2, -1, (M+m1+m2)*Dt^2], |
---|
1138 | [m1*L^2*Dt^2-m1*L*g, 0, 0, m1*L*Dt^2], |
---|
1139 | [0, m2*L^2*Dt^2-m2*L*g, 0, m2*L*Dt^2]; |
---|
1140 | module R = transpose(RR); |
---|
1141 | list C = control(R); |
---|
1142 | list CC = canonize(C); |
---|
1143 | view(CC); |
---|
1144 | } |
---|
1145 | //--------------------------------------------------------------- |
---|
1146 | static proc smdeg(matrix N) |
---|
1147 | // returns an intvec of length 2 with the index of an element of N with smallest degree |
---|
1148 | { |
---|
1149 | int n = nrows(N); |
---|
1150 | int m = ncols(N); |
---|
1151 | int d,d_temp; |
---|
1152 | intvec v; |
---|
1153 | int i,j; // counter |
---|
1154 | |
---|
1155 | if (N==0) |
---|
1156 | { |
---|
1157 | v = 1,1; |
---|
1158 | return(v); |
---|
1159 | } |
---|
1160 | |
---|
1161 | for (i=1; i<=n; i++) |
---|
1162 | // hier wird ein Element ausgewaehlt(!=0) und mit dessen Grad gestartet |
---|
1163 | { |
---|
1164 | for (j=1; j<=m; j++) |
---|
1165 | { |
---|
1166 | if( deg(N[i,j])!=-1 ) |
---|
1167 | { |
---|
1168 | d=deg(N[i,j]); |
---|
1169 | break; |
---|
1170 | } |
---|
1171 | } |
---|
1172 | if (d != -1) |
---|
1173 | { |
---|
1174 | break; |
---|
1175 | } |
---|
1176 | } |
---|
1177 | for(i=1; i<=n; i++) |
---|
1178 | { |
---|
1179 | for(j=1; j<=m; j++) |
---|
1180 | { |
---|
1181 | d_temp = deg(N[i,j]); |
---|
1182 | if ( (d_temp < d) && (N[i,j]!=0) ) |
---|
1183 | { |
---|
1184 | d=d_temp; |
---|
1185 | } |
---|
1186 | } |
---|
1187 | } |
---|
1188 | for (i=1; i<=n; i++) |
---|
1189 | { |
---|
1190 | for (j=1; j<=m;j++) |
---|
1191 | { |
---|
1192 | if ( (deg(N[i,j]) == d) && (N[i,j]!=0) ) |
---|
1193 | { |
---|
1194 | v = i,j; |
---|
1195 | return(v); |
---|
1196 | } |
---|
1197 | } |
---|
1198 | } |
---|
1199 | } |
---|
1200 | //--------------------------------------------------------------- |
---|
1201 | static proc NoNon0Pol(vector v) |
---|
1202 | // returns 1, if there is only one non-zero element in v and 0 else |
---|
1203 | { |
---|
1204 | int i,j; |
---|
1205 | int n = nrows(v); |
---|
1206 | for( j=1; j<=n; j++) |
---|
1207 | { |
---|
1208 | if (v[j] != 0) |
---|
1209 | { |
---|
1210 | i++; |
---|
1211 | } |
---|
1212 | } |
---|
1213 | if ( i!=1 ) |
---|
1214 | { |
---|
1215 | i=0; |
---|
1216 | } |
---|
1217 | return(i); |
---|
1218 | } |
---|
1219 | //--------------------------------------------------------------- |
---|
1220 | static proc Our_extgcd(poly p, poly q) |
---|
1221 | { |
---|
1222 | ideal J; //for extgcd-computations |
---|
1223 | matrix T; //----------"------------ |
---|
1224 | list L; |
---|
1225 | // the extgcd-command has a bug in versions before 2-0-7 |
---|
1226 | if ( system("version")<=2006 ) |
---|
1227 | { |
---|
1228 | J = p,q; // J = N[k-1,k-1],N[k,k]; //J is of type ideal |
---|
1229 | L[1] = liftstd(J,T); //T is of type matrix |
---|
1230 | if(J[1]==p) //this is just for the case the SINGULAR swaps the |
---|
1231 | // two elements due to ordering |
---|
1232 | { |
---|
1233 | L[2] = T[1,1]; |
---|
1234 | L[3] = T[2,1]; |
---|
1235 | } |
---|
1236 | else |
---|
1237 | { |
---|
1238 | L[2] = T[2,1]; |
---|
1239 | L[3] = T[1,1]; |
---|
1240 | } |
---|
1241 | } |
---|
1242 | else |
---|
1243 | { |
---|
1244 | L=extgcd(p,q); |
---|
1245 | // L=extgcd(N[k-1,k-1],N[k,k]); |
---|
1246 | //one can use this line if extgcd-bug is fixed |
---|
1247 | } |
---|
1248 | return(L); |
---|
1249 | } |
---|
1250 | //--------------------------------------------------------------- |
---|
1251 | proc smith( module M ) |
---|
1252 | "USAGE: smith(M), M a module or a matrix, |
---|
1253 | PURPOSE: computes the Smith form of a matrix |
---|
1254 | RETURN: a list of length 4 with the following entries: |
---|
1255 | @* [1]: The Smith-Form S of M, |
---|
1256 | @* [2]: the rank of M, |
---|
1257 | @* [3]: a unimodular matrix U, |
---|
1258 | @* [4]: a unimodular matrix V, |
---|
1259 | such that U*M*V=S. An empty list is returned when no Smith Form exists. |
---|
1260 | NOTE: The Smith form only exists over PIDs (principal ideal domains). |
---|
1261 | " |
---|
1262 | { |
---|
1263 | if (nvars(basering)>1) //if more than one variable, return empty list |
---|
1264 | { |
---|
1265 | list @L; |
---|
1266 | return (@L); |
---|
1267 | } |
---|
1268 | matrix N = matrix(M); //Typecasting |
---|
1269 | int n = nrows(N); |
---|
1270 | int m = ncols(N); |
---|
1271 | matrix P = unitmat(n); //left transformation matrix |
---|
1272 | matrix Q = unitmat(m); //right transformation matrix |
---|
1273 | int k, i, j, deg_temp; |
---|
1274 | poly tmp; |
---|
1275 | vector v; |
---|
1276 | list L; //for extgcd-computation |
---|
1277 | intmat f[1][n]; //to save degrees |
---|
1278 | matrix lambda[1][n]; //to save leadcoefficients |
---|
1279 | intmat g[1][m]; //to save degrees |
---|
1280 | matrix mu[1][m]; //to save leadcoefficients |
---|
1281 | int ii; //counter |
---|
1282 | |
---|
1283 | while ((k!=n) && (k!=m) ) |
---|
1284 | { |
---|
1285 | k++; |
---|
1286 | while ((k<=n) && (k<=m)) //outer while-loop for column-operations |
---|
1287 | { |
---|
1288 | while(k<=m ) //inner while-loop for row-operations |
---|
1289 | { |
---|
1290 | if( (n>m) && (k < n) && (k<m)) |
---|
1291 | { |
---|
1292 | if( simplify((ideal(submat(N,k+1..n,k+1..m))),2)== 0) |
---|
1293 | { |
---|
1294 | return(N,k-1,P,Q); |
---|
1295 | } |
---|
1296 | } |
---|
1297 | i,j = smdeg(submat(N,k..n,k..m)); //choose smallest degree in the remaining submatrix |
---|
1298 | i=i+(k-1); //indices adjusted to the whole matrix |
---|
1299 | j=j+(k-1); |
---|
1300 | if(i!=k) //take the element with smallest degree in the first position |
---|
1301 | { |
---|
1302 | N=permrow(N,i,k); |
---|
1303 | P=permrow(P,i,k); |
---|
1304 | } |
---|
1305 | if(j!=k) |
---|
1306 | { |
---|
1307 | N=permcol(N,j,k); |
---|
1308 | Q=permcol(Q,j,k); |
---|
1309 | } |
---|
1310 | if(NoNon0Pol(N[k])==1) |
---|
1311 | { |
---|
1312 | break; |
---|
1313 | } |
---|
1314 | tmp=leadcoef(N[k,k]); |
---|
1315 | deg_temp=ord(N[k,k]); //ord outputs the leading degree of N[k,k] |
---|
1316 | for(ii=k+1;ii<=n;ii++) |
---|
1317 | { |
---|
1318 | lambda[1,ii]=leadcoef(N[ii,k])/tmp; |
---|
1319 | f[1,ii]=deg(N[ii,k])-deg_temp; |
---|
1320 | } |
---|
1321 | for(ii=k+1;ii<=n;ii++) |
---|
1322 | { |
---|
1323 | N = addrow(N,k,-lambda[1,ii]*var(1)^f[1,ii],ii); |
---|
1324 | N = normalize(N); |
---|
1325 | P = addrow(P,k,-lambda[1,ii]*var(1)^f[1,ii],ii); |
---|
1326 | } |
---|
1327 | } |
---|
1328 | if (k>n) |
---|
1329 | { |
---|
1330 | break; |
---|
1331 | } |
---|
1332 | if(NoNon0Pol(transpose(N)[k])==1) |
---|
1333 | { |
---|
1334 | break; |
---|
1335 | } |
---|
1336 | tmp=leadcoef(N[k,k]); |
---|
1337 | deg_temp=ord(N[k,k]); //ord outputs the leading degree of N[k][k] |
---|
1338 | |
---|
1339 | for(ii=k+1;ii<=m;ii++) |
---|
1340 | { |
---|
1341 | mu[1,ii]=leadcoef(N[k,ii])/tmp; |
---|
1342 | g[1,ii]=deg(N[k,ii])-deg_temp; |
---|
1343 | } |
---|
1344 | for(ii=k+1;ii<=m;ii++) |
---|
1345 | { |
---|
1346 | N=addcol(N,k,-mu[1,ii]*var(1)^g[1,ii],ii); |
---|
1347 | N=normalize(N); |
---|
1348 | Q=addcol(Q,k,-mu[1,ii]*var(1)^g[1,ii],ii); |
---|
1349 | } |
---|
1350 | } |
---|
1351 | if( (k!=1) && (k<n) && (k<m) ) |
---|
1352 | { |
---|
1353 | L = Our_extgcd(N[k-1,k-1],N[k,k]); |
---|
1354 | if ( N[k-1,k-1]!=L[1] ) //means that N[k-1,k-1] is not a divisor of N[k,k] |
---|
1355 | { |
---|
1356 | N=addrow(N,k-1,L[2],k); |
---|
1357 | normalize(N); |
---|
1358 | P=addrow(P,k-1,L[2],k); |
---|
1359 | |
---|
1360 | N=addcol(N,k,-L[3],k-1); |
---|
1361 | N=normalize(N); |
---|
1362 | Q=addcol(Q,k,-L[3],k-1); |
---|
1363 | k=k-2; |
---|
1364 | } |
---|
1365 | } |
---|
1366 | } |
---|
1367 | if( (k<=n) && (k<=m) ) |
---|
1368 | { |
---|
1369 | if( N[k,k]==0) |
---|
1370 | { |
---|
1371 | return(N,k-1,P,Q); |
---|
1372 | } |
---|
1373 | } |
---|
1374 | return(N,k,P,Q); |
---|
1375 | } |
---|
1376 | example |
---|
1377 | { |
---|
1378 | "EXAMPLE:";echo = 2; |
---|
1379 | option(redSB); |
---|
1380 | option(redTail); |
---|
1381 | ring r=0,x,dp; |
---|
1382 | // see what happens when the matrix is already in Smith-Form |
---|
1383 | module M = [x,0,0],[0,x2,0],[0,0,x3]; |
---|
1384 | print(M); |
---|
1385 | list L = smith(M); |
---|
1386 | print(L[1]); |
---|
1387 | matrix N=matrix(M); |
---|
1388 | matrix B=L[3]*N*L[4]; |
---|
1389 | B=normalize(std(B)); |
---|
1390 | print(B); |
---|
1391 | //------- and yet another example -------------- |
---|
1392 | module M2=[x2,x,3x3-4],[2x2-1,4x,5x2],[2x5,3x,4x]; |
---|
1393 | print(M2); |
---|
1394 | list P=smith(M2); |
---|
1395 | print(P[1]); |
---|
1396 | matrix N2=matrix(M2); |
---|
1397 | matrix B2=P[3]*N2*P[4]; |
---|
1398 | print(B2); |
---|
1399 | B2=normalize(std(B2)); |
---|
1400 | print(B2); |
---|
1401 | } |
---|
1402 | //--------------------------------------------------------------- |
---|
1403 | proc list_tex(L, string name,link l,int nr_loop) |
---|
1404 | "USAGE: list_tex(L,name,l), where L is a list, name a string, l a link |
---|
1405 | writes the content of list L in a tex-file 'name' |
---|
1406 | RETURN: nothing |
---|
1407 | " |
---|
1408 | { |
---|
1409 | if(typeof(L)!="list") //in case L is not a list |
---|
1410 | { |
---|
1411 | texobj(name,L); |
---|
1412 | } |
---|
1413 | if(size(L)==0) |
---|
1414 | { |
---|
1415 | } |
---|
1416 | else |
---|
1417 | { |
---|
1418 | string t; |
---|
1419 | for (int i=1;i<=size(L);i++) |
---|
1420 | { |
---|
1421 | while(1) |
---|
1422 | { |
---|
1423 | if(typeof(L[i])=="string") //Fehler hier fuer normalen output->nur wenn string in liste dann verbatim |
---|
1424 | { |
---|
1425 | t=L[i]; |
---|
1426 | if(nr_loop==1) |
---|
1427 | { |
---|
1428 | write(l,"\\begin\{center\}"); |
---|
1429 | write(l,"\\begin\{verbatim\}"); |
---|
1430 | } |
---|
1431 | write(l,t); |
---|
1432 | if(nr_loop==0) |
---|
1433 | { |
---|
1434 | write(l,"\\par"); |
---|
1435 | } |
---|
1436 | if(nr_loop==1) |
---|
1437 | { |
---|
1438 | write(l,"\\end\{verbatim\}"); |
---|
1439 | write(l,"\\end\{center\}"); |
---|
1440 | } |
---|
1441 | break; |
---|
1442 | } |
---|
1443 | if(typeof(L[i])=="module") |
---|
1444 | { |
---|
1445 | texobj(name,matrix(L[i])); |
---|
1446 | break; |
---|
1447 | } |
---|
1448 | if(typeof(L[i])=="list") |
---|
1449 | { |
---|
1450 | list_tex(L[i],name,l,1); |
---|
1451 | break; |
---|
1452 | } |
---|
1453 | write(l,"\\begin\{center\}"); |
---|
1454 | texobj(name,L[i]); |
---|
1455 | write(l,"\\end\{center\}"); |
---|
1456 | write(l,"\\par"); |
---|
1457 | break; |
---|
1458 | } |
---|
1459 | } |
---|
1460 | } |
---|
1461 | } |
---|
1462 | //--------------------------------------------------------------- |
---|
1463 | proc verbatim_tex(string s, link l) |
---|
1464 | "USAGE: verbatim_tex(s,l), where s is a string and l a link |
---|
1465 | PURPOSE: writes the content of s in verbatim-environment in the file |
---|
1466 | specified by link |
---|
1467 | RETURN: nothing |
---|
1468 | " |
---|
1469 | { |
---|
1470 | write(l,"\\begin{verbatim}"); |
---|
1471 | write(l,s); |
---|
1472 | write(l,"\\end{verbatim}"); |
---|
1473 | write(l,"\\par"); |
---|
1474 | } |
---|
1475 | //--------------------------------------------------------------- |
---|
1476 | proc FindTorsion(module R, ideal TAnn) |
---|
1477 | "USAGE: FindTorsion(R, I); R an ideal/matrix/module, I an ideal |
---|
1478 | PURPOSE: computes the Groebner basis of the submodule of R, annihilated by I |
---|
1479 | RETURN: module |
---|
1480 | NOTE: especially helpful, when I is the annihilator of the t(R) - the torsion submodule of R. In this case, the result is the explicit presentation of t(R) as |
---|
1481 | the submodule of R |
---|
1482 | EXAMPLE: example FindTorsion; shows an example |
---|
1483 | " |
---|
1484 | { |
---|
1485 | // motivation: let R be a module, |
---|
1486 | // TAnn is the annihilator of t(R)\subset R |
---|
1487 | // compute the generators of t(R) explicitly |
---|
1488 | ideal AS = TAnn; |
---|
1489 | module S = R; |
---|
1490 | if (attrib(S,"isSB")<>1) |
---|
1491 | { |
---|
1492 | S = std(S); |
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1493 | } |
---|
1494 | if (attrib(AS,"isSB")<>1) |
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1495 | { |
---|
1496 | AS = std(AS); |
---|
1497 | } |
---|
1498 | int nc = ncols(S); |
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1499 | module To = quotient(S,AS); |
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1500 | To = std(NF(To,S)); |
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1501 | return(To); |
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1502 | } |
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1503 | example |
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1504 | { |
---|
1505 | "EXAMPLE:";echo = 2; |
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1506 | // Flexible Rod |
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1507 | ring A = 0,(D1, D2), (c,dp); |
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1508 | module R= [D1, -D1*D2, -1], [2*D1*D2, -D1-D1*D2^2, 0]; |
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1509 | module RR = transpose(R); |
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1510 | list L = control(RR); |
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1511 | // here, we have the annihilator: |
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1512 | ideal LAnn = D1; // = L[10] |
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1513 | module Tr = FindTorsion(RR,LAnn); |
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1514 | print(RR); // the module itself |
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1515 | print(Tr); // generators of the torsion submodule |
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1516 | } |
---|